Abstract: So far we have avoided to a large extent the more refined behavior of functions with respect to Lie derivatives. For the theory of spherical functions, we dealt with eigenvectors of convolution operators. The time has come to relate some invariants we have found in the representation theory with some of the invariant differential operators on G. Bargmann [Ba] saw how coefficient functions are eigenfunctions of such operators, Harish-Chandra got a complete insight into the situation by determining the center of the algebra of invariant differential operators, the centralizer of K in this algebra. Gelfand characterized spherical functions as eigenfunctions of this centralizer. In this chapter, we give Harish-Chandra’s result that there are no other spherical functions, besides those described in Chapter IV, on SL 2(R) where the proofs are short and easy.

Abstract: Since the conformation of physical systems is often advantageously described with the help of generalized (i.e. curvilinear) coordinates, the following questions are raised and general answers given to them: (i) What are the components of the momentum operators, those of the adjoint momentum operators and those of the hermitian momentum operators? (ii) What is the general form of the hamiltonian operator? (iii) What do the general forms of the momentum and hamiltonian operators become when expressed in terms of quasi-momentum operators (e.g. angular momentum component operators)? (iv) How are the answers to the above questions affected by an arbitrary choice of normalization convention for the total wave-function? A complete set of formulae (whatever the independent choices of coordinates, quasi-momentum operators and normalization) is given and various ‘historical’ formulae are shown to be particular instances of the general formulae proposed.

Abstract: I. Operators with Index.- 1. Fredholm Operators.- A. Hierarchy of Mathematical Objects.- B. Concept of Fredholm Operator.- 2. Algebraic Properties. Operators of Finite Rank.- A. The Snake Lemma.- B. Operators of Finite Rank and Fredholm Integral Equations.- 3. Analytic Methods. Compact Operators.- A. Analytic Methods.- B. The Adjoint Operator.- C. Compact Operators.- D. The Classical Integral Operators.- 4. The Fredholm Alternative.- A. The Riesz Lemma.- B. Sturm-Liouville Boundary-Value Problem.- 5. The Main Theorems.- A. The Calkin Algebra.- B. Perturbation Theory.- C. Homotopy-Invariance of the Index.- 6. Families of Invertible Operators. Kuiper's Theorem.- A. Homotopies of Operator-Valued Functions.- B. The Theorem of Kuiper.- 7. Families of Fredholm Operators. Index Bundles.- A. The Topology of F.- B. The Construction of Index Bundles.- C. The Theorem of Atiyah-Janich.- D. Homotopy and Unitary Equivalence.- 8. Fourier Series and Integrals (Fundamental Principles).- A. Fourier Series.- B. The Fourier Integral.- C. Higher Dimensional Fourier Integrals.- 9. Wiener-Hopf Operators.- A. The Reservoir of Examples of Fredholm Operators.- B. Origin and Fundamental Significance of Wiener-Hopf Operators.- C. The Characteristic Curve of a Wiener-Hopf Operator.- D. Wiener-Hopf Operators and Harmonic Analysis.- E. The Discrete Index Formula.- F. The Case of Systems.- G. The Continuous Analogue.- II. Analysis on Manifolds.- 1. Partial Differential Equations.- A. Linear Partial Differential Equations.- B. Elliptic Differential Equations.- C. Where Do Elliptic Differential Operators Arise?.- D. Boundary-Value Conditions.- E. Main Problems of Analysis and the Index Problem.- F. Numerical Aspects.- G. Elementary Examples.- 2. Differential Operators over Manifolds.- A. Motivation.- B. Differentiable Manifolds - Foundations.- C. Geometry of C? Mappings.- D. Integration on Manifolds.- E. Differential Operators on Manifolds.- F. Manifolds with Boundary.- 3. Pseudo-Differential Operators.- A. Motivation.- B. "Canonical" Pseudo-Differential Operators.- C. Pseudo-Differential Operators on Manifolds.- D. Approximation Theory for Pseudo-Differential Operators.- 4. Sobolev Spaces (Crash Course).- A. Motivation.- B. Definition.- C. The Main Theorems on Sobolev Spaces.- D. Case Studies.- 5. Elliptic Operators over Closed Manifolds.- A. Continuity of Pseudo-Differential Operators.- B. Elliptic Operators.- 6. Elliptic Boundary-Value Systems I (Differential Operators).- A. Differential Equations with Constant Coefficients.- B. Systems of Differential Equations with Constant Coefficients.- C. Variable Coefficients.- 7. Elliptic Differential Operators of First Order with Boundary Conditions.- A. The Topological Interpretation of Boundary-Value Conditions (Case Study).- B. Generalizations (Heuristic).- 8. Elliptic Boundary-Value Systems II (Survey).- A. The Poisson Principle.- B. The Green Algebra.- C. The Elliptic Case.- III. The Atiyah-Singer Index Formula.- 1. Introduction to Algebraic Topology.- A. Winding Numbers.- B. The Topology of the General Linear Group.- C. The Ring of Vector Bundles.- D. K-Theory with Compact Support.- E. Proof of the Periodicity Theorem of R. Bott.- 2. The Index Formula in the Euclidean Case.- A. Index Formula and Bott Periodicity.- B. The Difference Bundle of an Elliptic Operator.- C. The Index Formula.- 3. The Index Theorem for Closed Manifolds.- A. The Index Formula.- B. Comparison of the Proofs: The Cobordism Proof.- C. Comparison of the Proofs: The Imbedding Proof.- D. Comparison of the Proofs: The Heat Equation Proof.- 4. Applications (Survey).- A. Cohomological Formulation of the Index Formula.- B. The Case of Systems (Trivial Bundles).- C. Examples of Vanishing Index.- D. Euler Number and Signature.- E. Vector Fields on Manifolds.- F. Abelian Integrals and Riemann Surfaces.- G. The Theorem of Riemann-Roch-Hirzebruch.- H. The Index of Elliptic Boundary-Value Problems.- J. Real Operators.- K. The Lefsehetz Fixed-Point Formula.- L. Analysis on Symmetric Spaces.- M. Further Applications.- IV. The Index Formula and Gauge-Theoretical Physics.- 1. Physical Motivation and Overview.- A. Classical Field Theory.- B. Quantum Theory.- 2. Geometric Preliminaries.- A. Principal G-Bundles.- B. Connections and Curvature.- C. Equivariant Forms and Associated Bundles.- D. Gauge Transformations.- E. Curvature in Riemannian Geometry.- F. Bochner-Weitzenbock Formulas.- G. Chern Classes as Curvature Forms.- H. Holonomy.- 3. Gauge-Theoretic Instantons.- A. The Yang-Mills Functional.- B. Instantons on Euclidean 4-Space.- C. Linearization of the "Manifold" of Moduli of Self-Dual Connections.- D. Manifold Structure for Moduli of Self-Dual Connections.- E. Gauge-Theoretic Topology in Dimension Four.- Appendix: What are Vector Bundles?.- Literature.- Index of Notation Parts I, II, III.- IV.- Index of Names/Authors.

Abstract: A survey is given of the spectral properties of matrix finite-zone operators. Conditions of the type of J-self-adjointness for such operators and explicit formulas expressing the coefficients of such operators in terms of theta functions are obtained. The simplest examples of such J-self-adjoint, finite-zone operators turn out to be connected with the theory of ovals of plane, real, algebraic curves.

Abstract: In a recent manuscript [1] K-D. Kiirsten has produced counterexamples to two statements contained in our paper,, In the light of those results, we will discuss here at some length the appropriate modifications to the paper, deferring to a further publication [2] a detailed analysis (and generalization) of the counterexamples0 In the meantime a corrected version of our statements has been included in another work by one of us [3], The present addition to our paper results from extensive discusions, both orally and by correspondance, with Dr0 Kiirsten, Prof. G. Lassner and Dr. F. Mathot. We express our gratitude to all of them,,

Abstract: A class of pseufodifferential operators of infinite order acting on spaces of Gevrey functions and their duals is defined. For the corresponding symbols the rules of the classical symbolic calculus are proved. In particular for operators satisfying an “hipoellipticity condition” a result of propagation of Gevrey regularity, is obtained by proving the existence of a parametrix.

Abstract: In this chapter the Spectral Theorem for normal operators on a Hilbert space is proved. This theorem is then used to answer a number of questions concerning normal operators. In fact, the Spectral Theorem can be used to answer essentially every question about normal operators.

Abstract: Bloch Functions: The Basic Theory.- A Survey of Some Results on Subnormal Operators.- Optimization, Engineering, and a More General Corona Theorem.- Minimal Factorization, Linear Systems and Integral Operators.- Ha-Plitz Operators: A Survey of Some Recent Results.- Stochastic Processes, Infinitesimal Generators and Function Theory.- Paracommutators and Minimal Spaces.- Decomposition Theorems for Bergman Spaces and their Applications.- Operator-Theoretic Aspects of the Nevanlinna-Pick Interpolation Problem.- Cyclic Vectors in Banach Spaces of Analytic Functions.- Interpolation by Analytic Matrix Functions.

Abstract: Fuchsian differential operators with one singular point.- Lieear ordinary differential operators with one singular point.- Linear ordinary differential operators with several singular points.- Elliptic differential operators in ?n degenerating at one point.- Degenerate pseudodifferential operators on a closed curve.- A finite element method for pseudodifferential equations on a closed curve.- Appendix. Suboptimal convergence of the Galerkin method with splines for elliptic pseudodifferential equations.

Abstract: We study integral transforms of functions belonging to the Jakubowski class S(m, M) and determine the range of values of the exponent for which the integral is a convex or a close to convex function.

Abstract: The abstract abelian operator theory is developed from a general standpoint, using the method of forcing and Boolean-valued models. 1. Introduction. One aspect of the study of operator algebras is the description of the algebraic structure of algebras of operators, and representation of abstract algebras on a Hilbert space. This "algebraization" of the theory of algebras of operators is well understood in the case of bounded normal operators. The theory of von Neumann algebras (or the more general C *-algebras) is based on Stone's characterization of abelian (commutative) algebras of bounded operators in (13). Stone's theory describes such algebras axiomatically, in algebraic terms, without reference to Hilbert space, and develops a function calculus and the spectral theory for the abstract algebras. Moreover, the algebras are algebraically isomorphic to the algebra C(X) of all (complex-valued) continuous functions on an extremally disconnected compact Hausdorff space X. The functional representation of abelian von Neumann algebras has been used to extend Stone's work to abelian algebras of unbounded normal operators. This has been done for instance by Fell and Kelley in (2); a detailed account of spectral theory based on such an approach is presented by Kadison and Ringrose in (5). For a given abelian von Neumann algebra _, one defines an algebra _ of normal (not necessarily bounded) operators affiliated with -d. If C(X) is the functional algebra isomorphic to X, then -is isomorphic to the algebra of all normal functions on X. This provides both the spectral theory and a Borel function calculus for unbounded normal operators.

Abstract: Strong similarities between control theory and the theory on the solution of operator equations have been observed and basic results in control theory have been derived from operator theory arguments. The purpose of this work is to use the underlying duality in order to develop analysis and synthesis techniques for nonlinear systems. As an example, controllers induced by the Newton method are introduced and the corresponding stability characteristics are studied. The concepts are demonstrated by applications to linear and nonlinear systems.

Abstract: Motivated by a question that naturally arises concerning certain nonlinear integral operators, we give an extension, to multilinear maps, of the Banach-Steinhaus principle of uniform boundedness for linear operators. Applications are considered, and of particular interest to us are operators H that, for some positive integer p , have the representation (H_{x})(t)=\int_{0}^{t} \cdots \int_{0}^{t} k (t, \tau_{1}, \cdots ,\tau_{p})x(\tau_{1})\cdots x(\tau_{p})d\tau_{1} \cdots d \tau_{p}, t \geq 0 for an arbitrary bounded (Lebesgue measurable) complex-valued function x on [0, \infty) , where the kernel k has certain very reasonable integrability properties. We show, using the extension mentioned above, that such operators (which play an important role in the theory of representation of nonlinear systems) have the basic property that whenever they take the set of bounded functions into itself, there is a positive constant c such that \parallel Hx \parallel \leq c\parallel x \parallel^{P} for all bounded x , where \parallel \cdot \parallel denotes the usual sup norm; this had been proved earlier only for p = 1 . Related results for much more general cases are also given.

Abstract: A recently proposed method to compute the eigenvalues of differential operators, based on an appropriate finite-dimensional matrix representation of the differential operator, is tested on several eigenvalue problems, in one and more dimensions. The results confirm the wide applicability of this technique.